Effect of functionally graded material on frictionally excited thermoelastic instability

Wear 266 (2009) 139–146

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Effect of functionally graded material on frictionally excited thermoelastic instability Seung Wook Lee, Yong Hoon Jang ∗ School of Mechanical Engineering, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul 120-749, Republic of Korea

a r t i c l e

i n f o

Article history: Received 3 August 2007 Received in revised form 24 May 2008 Accepted 9 June 2008 Available online 18 July 2008 Keywords: Thermoelatic instability Functionally graded materials Non-homogeneous parameter Critical speed Transient evolution of temperature Contact pressure perturbation

a b s t r a c t Thermoelastic instability (TEI) is investigated focusing on the effect of functionally graded materials (FGM) where the FGM half plane slides against a homogeneous conducting or rigid non-conducting body at speed V. Results confirm analytically that there is maximum critical speeds occurring at a certain value of nonhomogeneous parameter of FGM within a range of thermal conductivity, which was reported previously in the simulation model [Y.H. Jang, S.-H. Ahn, Frictionally-excited thermoelastic instability on functionally graded material, Wear 262 (2007) 1102–1112]. The effect of non-homogeneous parameters for FGM on thermoelastic instability shows that the non-homogeneous parameters of the elastic modulus and thermal expansion coefficient are strong influential factors. It is also found that when the non-homogeneous parameters of FGM range between two positive bounded values, unconditionally stable behavior is shown for a range of thermal conductivity. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Concentration of frictional heating over zones smaller than the nominal frictional interface can occur in clutches and brakes, thus leading to high localized temperature and mechanical pressure. This feedback process is generally found to be unstable and is called thermoelastic instability (TEI) [2]. The resulting high local temperatures and thermal stresses also have various undesirable effects such as material transformations, thermal cracking, brake fade and thermoelastic buckling of disks. Interest in these phenomena is increasing in the automotive industry, prompted by changes in brake materials and other design improvements, in particular towards reducing noise and increasing comfort. Since the first physical explanation of TEI proposed by Barber [2], research has continued to study this problem. A theoretical study of TEI was presented by Burton et al. [3], who introduced the idea of critical speed. Most of the theoretical works have focused on the analysis of the onset of instability and were dedicated to the estimation of the critical speed for different geometries and material properties (see [4–8]). Especially, Burton et al. [3] analyzed the cases of one non-conducting material and identical material, concluding for the latest case that the sliding system is unconditionally stable

∗ Corresponding author. Tel.: +82 2 2123 5812; fax: +82 2 312 2159. E-mail address: [email protected] (Y.H. Jang). 0043-1648/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2008.06.006

for realistic friction coefficients. This statement was then corrected by Lee and Barber [4] who showed that the system is unconditionally stable depending on the value of material parameters. More recently, Harshock et al. [9], predicted the the relative importance of material parameters on the variation of critical speed by considering a fixed percentage variation of rotor and break-pad materials with respect to a base configuration. New material investigation on TEI was done by Decuzzi and Demelio [10] who investigated TEI phenomenon for two different sets of mating materials, namely steel–paper based frictional material and carbon–carbon composites. Recently, new material investigations and more accurate theoretical models have been observed mostly because of the ever increasing performances of brakes and clutches used in automotive, railway and aerospace applications. As promising alternative materials, functionally graded materials (FGM) [11,12] defined as material featuring a smooth gradation between two or more nonhomogeneous materials have been proposed. Recent investigation to FGM on TEI has been started by Jang and Ahn [1], who performed a numerical simulation, showing that the maximum critical speed can be obtained when non-homogeneity reaches a certain value, where the thin coating of FGM on core steel material slides against the conventional frictional material. To understand clearly the effect of non-homogeneous parameter on TEI, we shall consider the thermoelastic half plane which is composed of functionally graded material sliding against a homogeneous conducting or rigid non-conducting body at speed V, to

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distribution is assumed to be an exponential form such as K1 = K0 eıy1 ,

c1 = c0 eε1 y1 ,

E1 = E0 eˇy1 ,

˛1 = ˛0 ey1

1 = 0ε2 y1 , (3)

where K1 , c1 , 1 , E1 , ˛1 are thermal conductivity, specific heat, density, Young’s modulus, coefficient of thermal expansion, respectively. In exponential function, ı, ε1 , ε2 , ˇ,  are non-zero non-homogeneous parameters of the thermal conductivity, specific heat, density, Young’s modulus, coefficient of thermal expansion, respectively. The material property with subscript “0” is denoted as a constant property. 2.2. Temperature field The temperature field is governed by the Fourier equation for the body (1) and the body (2) shown as

Fig. 1. A half plane (1) composed of FGM sliding with relative speed V against a half plane of homogeneous frictional material (2).

show that it permits us to find analytically the critical conditions for the onset of instability. We will investigate the effect of nonhomogeneous parameters of FGM on thermoelastic instability. 2. Model and formulation We consider the thermoelastic half plane (y < 0) which is composed of functionally graded material in body (1), sliding against a homogeneous half plane of body (2) (y > 0), at a velocity V, as shown in Fig. 1. The two bodies are pressed together by a uniform pressure p0 applied remotely to ensure complete contact along the interface. We assume that the properties of FGM body (1) are varied continuously with the function of y. The uniform contact pressure is perturbed superimposing a sinusoidal perturbation with wave number m at the sliding interface (y = 0) having the form p(x, t) = p0 ebt ejmx

(1)

where the growth rate b can be (i) negative—stable perturbation, (ii) positive—unstable perturbation, and (iii) zero—threshold of instability, for which the critical speed Vcr is determined. We assume that the bodies have absolute velocities Vi (i = 1, 2) in x-direction and that the perturbation develops in the uniform fields which has absolute velocity c and relative velocity ci with respect to body i. It follows that c = V1 + c1 = V2 + c2 and that the sliding velocity V can be written V = V2 − V1 = c1 − c2

(2)

∂2 T1 ∂T1 1 ∂T1 ∂2 T1 + +ı = 2 k1 ∂t ∂y1 ∂x1 ∂y1 2

(4)

∂2 T2 1 ∂T2 ∂2 T2 + = 2 k2 ∂t ∂x2 ∂y2 2

(5)

where ki = Ki /(i ci ) is the thermal diffusivity. Although the thermal diffusivity is also changed with location in FGM, we confine the non-homogeneous parameters of thermal conductivity, density and specific heat as a form of ı = ε1 + ε2 , obtaining a constant thermal diffusivity of k1 . This selection of material properties makes it possible for us to obtain an analytic solution. In body (2), the thermal diffusivity of k2 is a constant value. The temperature perturbations in each body which satisfy the transient heat conduction equation can be written as (6) Ti (x, y, t) = fi (yi ) ebt ejm(xi −ci t) √ where j = −1. Substituting Eq. (6) into Eq. (4) and satisfying that the temperature perturbation must decay away from the interface, we obtain the temperature perturbation for body (1) T1 (x1 , y1 , t) = F e1 y1 ebt ejm(x1 −c1 t) where



1 1 = 2

 ı2

−ı +

+4



m2

b − jmc1 + k1

(7)



 (8)

and F is an unknown constant. Following the same analysis procedure for body (1), we obtain the temperature perturbation of body (2) as T2 (x2 , y2 , t) = G e2 y2 ebt ejm(x2 −c2 t) where



2 = −

m2 +

b − jmc2 k2

(9)

(10)

The frame of reference stationary with bodies (1) and (2) are introduced, as in Fig. 1, where x = x1 − c1 t = x2 − c2 t and y = y1 = y2 . We shall investigate the stability of the system by finding the condition under which a small perturbation in the temperature and stress fields can grow with times.

and rewriting Eqs. (7) and (9) using the global coordinate, we obtain

2.1. Material property of the FGM

T1 (x, y, t) = T0 e1 y ebt ejmx

(12)

jmx

(13)

and G is also an unknown constant. A continuity condition is applied to temperature fields at the interface y = 0 as T1 (x, 0, t) = T2 (x, 0, t)

2 y

The FGM used in this analysis is assumed to be processed in such a way that the property grading is smooth. The material property is a function of y-coordinate. Specifically, the function of FGM

T2 (x, y, t) = T0 e

bt

e e

(11)

where T0 is an unknown constant which is determined by other boundary condition.

S.W. Lee, Y.H. Jang / Wear 266 (2009) 139–146

2.3. Thermoelastic stresses and displacements The governing equation for coupled thermoelasticity without body force is ∂ij ∂xj

=0

(14)

The general solutions of stress fields of body (1) in global coordinates are obtained by substituting Eqs. (23) and (24) into (20) and (21) 2

with the constitutive law

=

E , 2(1 + )

¯ = 2 ,  1 − 2

eij =

1 2



∂uj ∂ui + ∂xj ∂xi

∇ 2 ui +

2 Ri − 1

∂2 vi ∂2 ui + ∂xi ∂yi ∂xi 2





+ˇ

∂v ∂ui + i ∂yi ∂xi



=



4˛¯ i eyi ∂Ti Ri − 1 ∂xi (17)

∇ 2 vi + =

2 Ri − 1

4˛¯ i eyi Ri − 1

∂2 vi ∂2 ui + 2 ∂xi ∂yi ∂yi



(ˇ + )Ti +

 +

∂Ti ∂yi



ˇ Ri − 1

 (1 + Ri )

∂vi ∂u +(3 − Ri ) i ∂yi ∂xi

where 1 ri = 2



(19)

ˇyi

xyi = i e

∂v ∂u (1 + Ri ) i + (3 − Ri ) i − 4˛¯ i eyi Ti ∂xi ∂yi ∂ui ∂v + i ∂yi ∂xi



(20)



(21)

2.3.1. Body 1 The solutions of differential equations (17) and (18) are composed of homogeneous and particular solution. Introducing the displacement fields that vary sinusoidally parallel to the sliding surface vi (x, y) = Vi (y) ejmx

(i = 1, 2)

(22)

where U(y), V (y) are generally complex functions to describe the in-phase and out-of-phase components of displacement field. Two ordinary differential equations for U(y) and V (y) are obtained by substituting Eq. (22) into (17) and (18), resulting in the displacements in global coordinate as u1 (x, y) =

2  i=1



ri y

Ai e

e

jmx

4˛¯1 T0 + jm M1 e(+1 )y ebt+jmx R1 − 1

2

v1 (x, y) =

i=1

jsi Ai eri y ejmx +





 3 − R 1/2

i

−ˇ +

4m2 + ˇ2 + (−1) 4mˇj



1

R1 + 1

(27)

(28)

P − [Q + 2ˇ((R1 − 2)/(R1 − 1))][ˇ +  +

]

[((R1 − 1)/(R1 + 1))P − 4m2 R1 /(R12 − 1)]P + m2 [Q + 2ˇ((R1 − 2)/(R1 − 1))]Q (29)

P=

(ˇ +  +

4˛¯1 T0 M2 e(+1 )y ebt+jmx R1 − 1

(23)

)[((R1 − 1)/(R1 + 1))P − (4m2 R1 )/(R12 − 1)] + m2 Q

[((R1 − 1)/(R1 + 1))P − 4m2 R1 /(R12 − 1)]P + m2 [Q + 2ˇ((R1 − 2)/(R1 − 1))]Q (30)

R − 1 1

Q =ˇ

For body (2), the governing equations and the stress fields can be expressed by ˇ =  = 0.

ui (x, y) = Ui (y) ejmx ,

(26)

[2ri + (R1 − 1)ˇ]m

M1 =

for the plane strain state. The stresses are represented as yyi

(ri − msi )Ai eri y ejmx

(18)

Ri = 3 − 4 i



2 

(R1 − 1)(ri2 + ˇri ) − (R1 + 1)m2

M2 =

 eˇyi = i Ri − 1

(25)

[(1 + R1 )(1 + )M2

4˛¯1 T0 1 +jm [(1 + )M1 + M2 ] e(+1 +ˇ)y ebt+jmx R1 − 1



where ˛¯ i = ˛i (1 + i ),

i=1

i=1

(16)

si =



2

xy1 (x, y) = 1 eˇy

Eliminating the stresses between (14–16), we have the Navier equation for the displacement field



4˛¯1 T0 1

(R1 − 1) + (3 − R1 )M1 − (R1 − 1)] e(+1 +ˇ)y ebt+jmx

(15)

where

1 eˇy  j[(1 + R1 )si ri + (3 − R1 )m]Ai eri y ejmx R1 − 1

yy1 (x, y) = +

¯ ij ekk + 2eij − (3 ¯ + 2)˛ıij T ij = ı

141

R1 + 1

(

3 − R  1

R1 − 1

+ ) + ˇ

R1 + 1 ( R1 − 1

2 ( R1 − 1

+ )

2

+

where Ai (i = 1, 2) is the unknowns to be determined by boundary conditions. 2.3.2. Body 2 Following the same steps as body (1), we obtain the displacements which satisfy the regularity condition u2 (x, +∞) = 0, v2 (x, +∞) = 0, showing as u2 (x, y) = (B1 + B2 y) e−my ejmx + j





v2 (x, y) = j B1 + B2 y + −

R1 m



4˛¯2 m 22

− m2

T0 e−2 y ebt+jmx (33) R2 + 1

e−my ejmx (34)

4˛¯2 2 T0 e−2 y ebt+jmx 22 − m2 R2 + 1

The corresponding stresses are



yy2 (x, y) = 23 −jm(B1 + B2 y) + +

1 + R  2

m T0 −2 y bt+jmx e e 22 − m2 R2 + 1

−j

B2 e−my ejmx (35)

4˛¯2 m2



where the displacements are satisfied with the regularity condition, i.e. u1 (x, −∞) = 0, v1 (x, −∞) = 0.

(31) (32)

xy2 (x, y) = 23 −m(B1 + B2 y) + (24)

+ ) − m2

1 − R 

m T0 −2 y bt+jmx e e 22 − m2 R2 + 1

4˛¯2 m2

2

B2 e−my ejmx (36)

where Bi (i = 1, 2) is unknowns to be determined by boundary conditions.

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2.4. Boundary conditions This thermoelastic problem has five unknowns T0 , Ai , Bi (i = 1, 2) to be determined by the boundary conditions at the interface. Four equations are obtained from the mechanical system at the contact interface, y = 0, showing as v1 (x, 0) = v2 (x, 0)

(37)

yy1 (x, 0) = yy2 (x, 0)

(38)

xy1 (x, 0) = xy2 (x, 0)

(39)

xy1 (x, 0) = fyy1 (x, 0)

(40)

We also apply the regularity condition to stress fields since the stresses due to the temperature perturbation must disappear at a distance far from the interface, yy1 , xy1 → 0 at y → −∞, leading to



 3 − R 1/2

4m2 + ˇ2 + (−1)i 4mˇj

ˇ+



ˇ+ +

1 −ı + 2



1

R1 + 1



ı2 + 4 m2 − j

mc1 k1



>0

(41)

>0

(42)

The final conditions is to satisfy that the rate of heat generation at the interface due to the frictional tractions is equal to the rate of conduction away from the interface, i.e. ∂T1 ∂T2 (x, 0, t) + K2 (x, 0, t) = fVp(x, t) ∂y ∂y

qnet (x, 0, t) = −K1

(43)

1 1+ 2

2∗ =

 ∗2 = −





1/2

1 + (c2∗ )2

1 −1 + 2

(50)





1/2

1 + (c2∗ )2

(51)

where i has to be positive, because the perturbation amplitude has to decay away from the friction interface. Using these dimensionless stability boundaries and parameters, we can obtain a dimensionless characteristic equation and hence a critical speed, Vcr , by setting the growth rate to zero. ∗ y

−K ∗ T0 ∗1 e

1

∗ y

+ ∗2 e

2

=

1 ∗ ∗ fV yyi K2

(52)

∗ is the dimensionless form of  jmx . Since where yy yyi without e i Eq. (52) is complex equation, both the real and imaginary parts of the equation satisfy simultaneously. We do not show the real and imaginary part of Eq. (52) because of lengthy expressions.

2.6. Critical speed for non-conducting rigid half space The critical speed for the case where one of the bodies is a non-conducting rigid half space can be obtained from the general solution by a limiting process. Taking the non-conducting body to be body (2), we obtain



∗ Vcr

= Re

K1 ∗1

H1 4fk1 1 ˛1 (1 + 1 ) H2

(53)

2.5. Dimensionless form of the stability condition where We rewrite the characteristic equation (43) in dimensionless form by using following dimensionless parameters K∗ =

K1 , K2

k∗ =

∗i =

i , m

ci∗ =

ci , k2 m

ˇ∗ =

ˇ , m

ı∗ =

ı , m

k1 , k2

˛∗ =

˛1 (1 + 1 ) ˛2 (1 + 2 ) V = c1∗ − c2∗ k2 m

V∗ = ∗ =

(44)

H1 = (R1 − 1)(r2∗ s1∗ − r1∗ s2∗ ) +jf [(R1 − 3)(s1∗ − s2∗ ) + (R1 + 1)(r1∗ − r2∗ )s1∗ s2∗ ]

R1 − 3 ∗ ∗ {M [(s − s2∗ )( ∗ + ∗1 ) + r1∗ s2∗ − r2∗ s1∗ ] + M2∗ (r1∗ − r2∗ )} R1 − 1 1 1 R1 + 1 ! ∗ ∗ ∗ + M1 [s1 s2 (r1 − r2 )( ∗ + ∗1 )] (55) R1 − 1 " ∗ ∗ ∗ + M2 [(r1 s2 − r2∗ s1∗ )( ∗ + ∗1 ) + r1∗ r2∗ (s1∗ − s2∗ )] +(r2∗ s1∗ − r1∗ s2∗ )

H2 =

(45)

 m

(54)

(46)

The characteristic equation (43) serves to determine the exponential growth rate b for a disturbance of given frequency m, and sliding speed V. Perturbation will generally only be possible for certain eigenvalues of the exponential growth rate b. Stability will be maintained if the growth rates of all such perturbations are negative since all initial perturbations will then decay with time. Thus we can find the stability boundary by setting the growth rate to zero and hence obtain the critical sliding speed which depends upon the wavelength of the perturbation. When b = 0, the dimensionless forms of Eqs. (8) and (10) reduces to

and

∗i = i∗ + j∗i

The above formulation is employed in this section to estimate the influence of the non-homogeneous parameters in FGM on the ∗ for two cases: (i) the case of non-conducting rigid critical speed Vcr half space in body (2) and (ii) the case of homogeneous conducting deformable half space in body (2). Appropriate friction material and FGM properties, reproduced from Jang and Ahn [1] are given in Table 1.

1∗ =

1 2

⎧ ⎪ ⎨

(47)

⎡ ⎛

−ı∗ + ⎣

⎪ ⎩

1 2

⎝(ı∗2 + 4) +



 2

(ı∗2 + 4) +

4c1∗ k∗

2

⎞⎤1/2 ⎫ ⎪ ⎬ ⎠⎦ ⎪ ⎭ (48)

⎧⎡ ⎛ ⎞⎤1/2 ⎫  ⎪ ⎪  ∗ 2 ⎨ ⎬ 4c1 1 2 ∗ 1 ∗2 ∗2 ⎣ ⎝ ⎠ ⎦ − −(ı + 4)+ (ı + 4) + (49) 1 = ∗ 2⎪ k ⎪ ⎩ 2 ⎭

ri∗ =

ri , m

Q∗ =

Q m

si∗ = si M1∗ = m2 M1∗ ,

M2∗ = mM2∗ ,

P∗ =

P , m2 (56)

3. Results

3.1. Non-conducting rigid material For the non-conducting rigid material of body (2), we select steel material at the contact interface of FGM to obtain the validity of the current analysis. For this purpose, we define the ratio of the

S.W. Lee, Y.H. Jang / Wear 266 (2009) 139–146

143

Table 1 Material properties of FGM and frictional material Properties

Friction material

Steel

Ceramic

Elastic modulus (GN/m2 ) Thermal expansion (␮ ◦ C−1 ) Thermal conductivity (W/m ◦ C) Poisson’s ratio Thermal diffusivity (␮m/s2 )

0.3 14 0.241 0.12 0.177

200 12 42 0.3 11.9

151 12 3.0 0.3 1.15

critical speed obtained from the FGM in body (1) to that from the homogeneous material in body (1) [4] as RV =

∗ VFGM ∗ Vhomogeneous

(57)

where ∗ Vhomogeneous =

K1 (1 − 1 ) fk1 1 ˛1 (1 + 1 )

(58)

Fig. 2 shows the variation of RV according to the same non-homogeneous parameters of ˇ∗ ,  ∗ , ı∗ . When the nonhomogeneous parameters approach a value of zero, the ratio RV also approaches a value of one, meaning that the case of the FGM with the non-homogeneous parameters of zero is identical to the case of the homogeneous steel material. The ratio RV is increased monotonously as the non-homogeneous parameters increase. Specifically, when the non-homogeneous parameters is positive, the ratio RV is greater than 1, meaning that the critical speed of FGM is greater than that of homogeneous steel material of body (1). For the negative non-homogeneous parameter, the variation is reversed. Note that the shaded region of Fig. 1 represents the range of non-homogeneous parameter which do not satisfy the regularity condition of stress, i.e. Eqs. (41) and (42). Fig. 3 shows the relative critical speed of FGM with respect to that of the homogeneous steel for the non-homogeneous parameters of the thermal conductivity, the elastic modulus and the thermal expansion. In this calculation, we change a nonhomogeneous parameter from −1 to 1 while we set two parameters as a value of zero. It confirms that the non-homogeneous parameter of thermal conductivity exerts a strong influence on the variation of critical speed. The critical speed with respect to the non-homogeneous parameter of the elastic modulus reduces while the other non-homogeneous parameters increase.

Fig. 2. Ratio of critical speed of the FGM to that of the homogeneous material as a function of the non-homogeneous parameter of FGM for the model; body (1): FGM-steel, body (2): rigid, non-conducting material.

Fig. 3. Variation of the critical speed ratio as a function of the non-homogeneous parameters ˇ∗ , ı∗ ,  ∗ of FGM for the model; body (1): FGM-steel, body (2): rigid, non-conducting material.

3.2. Conducting elastic material When the frictional material of body (2) is a deformable conducting material, the behavior of critical speed is different and in some cases the system is stable for all speeds. Firstly, the material of FGM at contact interface is chosen to steel. Fig. 4 shows the critical speed according to the same nonhomogeneous parameters of ˇ∗ ,  ∗ , ı∗ for various ratio of thermal conductivity K ∗ . More interesting result is shown that when the ratio of thermal conductivity K ∗ is less than a certain value of 0.0057 which is the typical thermal conductivity ratio of steel and frictional material, the critical speeds are bounded and have a maximum value at a certain value of the non-homogeneous parameters. Above a certain value of thermal conductivity of 0.0057, the critical speed increases as the non-homogeneous parameters reduce. The phenomenon of the maximum critical speed occurring at a specific value of non-homogeneous parameters is also reported by Jang and Ahn [1] who simulated a numerical model of FGM layer and a frictional material to investigate the thermoelastic instability due to frictional heating.

∗ Fig. 4. Dimensionless critical speed Vcr against non-homogeneous parameters (ˇ∗ =  ∗ = ı∗ ) for different values of the thermal conductivity ratio K ∗ in the model; body (1): FGM-steel, body (2): conducting elastic material.

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S.W. Lee, Y.H. Jang / Wear 266 (2009) 139–146

Fig. 5. Variation of the critical speed ratio as a function of the non-homogeneous parameters ˇ∗ , ı∗ ,  ∗ of FGM for the model; body (1): FGM-steel, body (2): conducting elastic material.

Fig. 7. Variation of the critical speed ratio as a function of the non-homogeneous parameters ˇ∗ , ı∗ ,  ∗ of FGM for the model; body (1): FGM-ceramic, body (2): conducting elastic material.

Fig. 5 shows the relative critical speed of FGM with respect to that of the homogeneous steel according to the non-homogeneous parameters of the thermal conductivity, the elastic modulus and the thermal expansion. It shows that the non-homogeneous parameters of elastic modulus and thermal expansion coefficient are strong influential factors to the variation of critical speed while the thermal conductivity is less sensitive to the variation of critical speed. The critical speed with respect to the non-homogeneous parameter of the thermal conductivity reduces whereas the other non-homogeneous parameters increase. The effect of nonhomogeneous parameter on the critical speed is different from the case of non-conducting rigid frictional material as shown in Fig. 3. Secondly, a ceramic material is chosen for the material of the contact interface of FGM, ceramic material. Fig. 6 shows the critical speed according to the same non-homogeneous parameters of ˇ∗ ,  ∗ , ı∗ for various ratio of thermal conductivity K ∗ . As shown in Fig. 4, there is a non-homogeneous parameter which has the maximum critical speed within a certain range of the ratio of thermal conductivity, i.e. 0 < K ∗ ≤ 0.005. After K ∗ is greater than 0.006, the critical speed varies exponentially and eventually does not exist. We show the result of K ∗ = 0.8 which is the typical

thermal conductivity ratio of ceramic and frictional material, showing a obvious distinction of critical speed. To clearly evaluate this phenomenon, Fig. 7 is plotted for the variation of critical speed according to the ratio of thermal conductivity for the several values of non-homogeneous parameters. When the non-homogeneous parameters ˇ∗ =  ∗ = ı∗ is less than the value of 13, there is no critical speed since the critical speed is exceptionally high, meaning that the system is stable for all speeds, i.e. implying that most cases involving similar materials will be unconditionally stable. A similar behavior of unconditional stable system was reported by Lee and Barber [4] who investigated a model of homogeneous steel and frictional materials. 3.3. Transient evolution of the temperature perturbation amplitude We investigate the effect of the FGM layer on the transient evolution of temperature perturbation. According to Al-Shabibi and Barber [13] and Afferrante et al. [14], as long as full contact is retained, the thermoelastic contact problem is linear, and an alternative to a time-marching scheme is to expand the temperature field as an eigenfunction series. If the sliding speed is a constraint, each term in the series evolves according to its own growth rate and the full transient solution can be written down in terms of the initial conditions. Thus the temperature and the corresponding pressure perturbation development in the transient range can be written as

#

t

T = T (0) exp

#

b(V (t)) dt

(59)

b(V (t)) dt

(60)

0

p = p(0) exp

t

0

where T (0) and p(0) are the temperature and the contact pressure at the time t = 0 and b(t) is the dominant growth rate at each speed V (t). Thus, if we know the variations of b and V from the stability equation and assume that during the frictional engagement process the sliding speed V reduces linearly with time



V (t) = V0 1 − ∗ Fig. 6. Dimensionless critical speed Vcr against non-homogeneous parameter (ˇ∗ =  ∗ = ı∗ ) for different values of the thermal conductivity ratio K ∗ in the model; body (1): FGM-ceramic, body (2): conducting elastic material.

t t0



(61)

where V0 and t0 are the initial sliding speed and stopping time, respectively, the transient evolution of the tempera-

S.W. Lee, Y.H. Jang / Wear 266 (2009) 139–146

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vantage of the piecewise homogeneous material, which leads to discontinuities in the material’s mechanical, physical and chemical properties along the interfaces and consequently higher residual, thermal and mechanical stresses, weaker bonding strength, low toughness and tendency toward cracking and spallation. Thus, FGM has an advantage that accrues from a graded disc which enables one to keep the same tribological sliding pair, whilst using a material with different properties elsewhere, for example, to increase conductivity and hence improve heat transfer in the brake. What we add one more technical things obtained from the current results is that when the materials of sliding pair are chosen, the highest critical speed and hence the best judder performance will be obtained if the transition layer is approximately thick, presumably related to other disc dimensions, and the gradient of material grading is large. Therefore, FGM is better than a homogeneous disc in improving the performances of actual brakes and friction systems. Fig. 8. Transient evolution of the temperature and pressure perturbation normalized with respect to its initial value T (0) and p(0) for FGMs having steel or ceramic material at the interface when ˇ∗ =  ∗ = ı∗ = 5, respectively. The dashed and solid lines represent the case of steel- and ceramic-FGM, respectively.

ture perturbation amplitude at the sliding interface can be derived. According to the variation of non-homogeneous parameters of the FGM having steel or ceramic material at the interface, the evolution over time of the temperature amplitude normalized with respect to its initial value T0 and p0 is shown in Fig. 8. The specific non-homogeneous parameters (ˇ∗ = ı∗ =  ∗ ) of 5 is fixed. The FGM having ceramic material at the interface has lower temperature perturbation amplitude during the evolution of engagement time than that of steel material at the interface. As pointed out in Fig. 6, the FGM having ceramic material at the interface have a better performance on TEI of reducing the susceptibility towards hot spotting and the severity of thermomechanical damage. 3.4. Implication for brake disc design The current study is intended to investigate the effect of nonhomogeneous parameter of FGM on TEI, particulary focusing on the critical speed. We have also suggested reference results by introducing the solution of TEI for the sliding half plane. We believe that the results would be utilized appropriately for the design of any frictional sliding system such as brakes or clutches under a limited condition. FGM, which has its own characteristics of wear resistance and thermal barrier, is generally possible to apply to the frictional sliding system. For example, FGM is applied to the disc of brake to improve the performance of brake. The important factors determining the performance of brake are the properties of the surface layer which will be dictated by tribological considerations such as friction coefficient and wear rate. Several materials of conventional brake disc are steel or cast iron and aluminum because of the better conductivity. The thermomechanical performance of these disc materials can be enhanced by borrowing an appropriate coating layer. We used a ceramic material at the frictional surface. Ceramic material, which is already used in many industrial applications such as cutting tools, gears, bearing and cams, provides the necessary hardness and wear resistance to surface of structural components transmitting forces through contact. In addition, ceramic based on oxides such as Al2 O3 are well-known and used in continuous turning of case iron and steel. By adding TiC, it is possible to improve thermal stability [15]. Furthermore, by grading continuously the surface material to steel or cast iron, we can eliminate the disad-

4. Conclusion The analytical model is investigated for the thermoelastic instability due to frictional heating where the FGM half plane slides against a rigid non-conducting or a homogeneous conducting body at speed V. The results of the analysis is validated by confirming the case of homogeneous steel and rigid non-conducting body. The above results demonstrates analytically that there is a maximum critical speed occurring at a certain value of non-homogeneous parameter within the range of thermal conductivity. The material sensitivity to the critical speed for the non-homogeneous parameter of FGM contacting a homogeneous frictional material shows that the non-homogeneous parameters of the elastic modulus and thermal expansion coefficient are strong influential factors while the thermal conductivity is less sensitive to the variation of critical speed. This effect of non-homogeneous parameter on the critical speed is different from the case of rigid, non-conducting frictional material. In the special case where the non-homogeneous parameters of FGM range between two positive bounded values, unconditionally stable behavior is predicted for a range of thermal conductivity. We finally obtain the transient evolution of temperature and pressure perturbation of FGM having steel or ceramic meterial at the interface, showing better performance of ceramic FGM on TEI. By applying the current methodology and results to a finite FGM layer model for thermoelastic instability problem, a practical criterion can be sought to evaluate the susceptibility of instability in frictional heating in brake or clutch structure. Acknowledgment The authors wish to acknowledge Prof. J. R. Barber of the University of Michigan for his helpful comments. References [1] Y.H. Jang, S-H. Ahn, Frictionally-excited thermoelastic instability on functionally graded material, Wear 262 (2007) 1102–1112. [2] J.R. Barber, Thermoelastic instabilities in the sliding of conforming solids, Proc. Roy. Soc. London A 312 (1969) 381–394. [3] R.A. Burton, V. Nerlikar, S.R. Kilaparti, Thermoelastic instability in a seal-like configuration, Wear 24 (1973) 177–188. [4] K. Lee, J.R. Barber, The effect of shear traction on frictionally-excited thermoelastic instability, Wear 160 (1993) 237–242. [5] K. Lee, J.R. Barber, Frictionally excited thermoelastic instability in automotive disk brakes, ASME J. Tribol. 115 (1993) 605–614. [6] Y.-B. Yi, J.R. Barber, P. Zagrodzki, Eigenvalue solution of thermoelastic instability problems using Fourier reduction, Proc. Roy. Soc. London Ser. A 456 (2000) 2799–2821. [7] P. Decuzzi, Frictionally excited thermoelastic instability in viscoelastic and poroelastic media, Int. J. Mech. Sci. 44 (2002) 585–600.

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[8] C.L. Davis, C.M. Krousgrill, F. Sadeghi, Effect of temperature on thermoelastic instability in thin disks, J. Tribol. 124 (2002) 429–437. [9] D.L. Hartsock, R.L. Hecht, J.W. Fash, Parametric analyses of thermoelastic instability in disk brakes, Int. J. Vehicle Design 21 (4/5) (1999) 510– 526. [10] P. Decuzzi, G. Demelio, The effect of material properties on the thermoelastic stability of sliding systems, Wear 252 (2002) 311–321. [11] M. Yamanouchi, T. Koisumi, L. Hirai, I. Shiota (Eds.), FGM-90, Proceedings of the First International Symposium on Functionally Gradient Materials, FGM Forum, Tokyo, 1990.

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